Prove that (cosec θ - θ)² = (1 - θ)/(1 + θ).

LHS = RHS = (1-θ)/(1+θ) (proved)

MathsCBSE Class 10 · Basic · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic sample paper 2023-24: question 30(a)

From the official CBSE sample question paper for 2023-24, section C, question 30(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(a) · 3 marks · Short answerIntroduction to Trigonometry

Prove that $(cosec θ - cot θ)^{2} = \frac{1 - c o s θ}{1 + c o s θ}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

Open one hint at a time and try again before the next.

Hint 1

Convert cosec θ and cot θ into sin θ and cos θ.

Hint 2

After squaring, write $sin^{2} θ$ as $1 - cos^{2} θ$ and factorise it as a difference of squares.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

LHS $= (cosec θ - cot θ)^{2} = (\frac{1}{s i n θ} - \frac{c o s θ}{s i n θ})^{2}$
½ mark
$= (\frac{1 - c o s θ}{s i n θ})^{2} = \frac{( 1 - c o s θ ) ^{2}}{s i n ^{2} θ}$
½ mark
$= \frac{( 1 - c o s θ ) ^{2}}{1 - c o s ^{2} θ}$ [since $sin^{2} θ = 1 - cos^{2} θ$ ]
1 mark
$= \frac{( 1 - c o s θ ) ^{2}}{( 1 - c o s θ ) ( 1 + c o s θ )} = \frac{1 - c o s θ}{1 + c o s θ} =$ RHS. Hence proved.
1 mark

Answer: LHS = RHS $= \frac{1 - c o s θ}{1 + c o s θ}$ (proved)

Where the marks are

Writes cosec and cot in terms of sin and cos (1/2 mark)
Combines to $\frac{( 1 - c o s θ ) ^{2}}{s i n ^{2} θ}$ (1/2 mark)
Replaces $sin^{2} θ$ by $1 - cos^{2} θ$ (1 mark)
Factorises, cancels and reaches the RHS (1 mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Proving Simple Trigonometric Identities

Practise this chapterBoard-style questions from Introduction to Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2023-24: question 30(a)

Hints

Step-by-step solution

Revise the chapter

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